Properties

Label 1089.6.a.p
Level $1089$
Weight $6$
Character orbit 1089.a
Self dual yes
Analytic conductor $174.658$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(174.657979776\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 7 - \beta ) q^{2} + ( 25 - 13 \beta ) q^{4} + ( -24 - 10 \beta ) q^{5} + ( -42 - 62 \beta ) q^{7} + ( 55 - 71 \beta ) q^{8} +O(q^{10})\) \( q + ( 7 - \beta ) q^{2} + ( 25 - 13 \beta ) q^{4} + ( -24 - 10 \beta ) q^{5} + ( -42 - 62 \beta ) q^{7} + ( 55 - 71 \beta ) q^{8} + ( -88 - 36 \beta ) q^{10} + ( 28 + 74 \beta ) q^{13} + ( 202 - 330 \beta ) q^{14} + ( 153 - 65 \beta ) q^{16} + ( -550 + 372 \beta ) q^{17} + ( -12 + 852 \beta ) q^{19} + ( 440 + 192 \beta ) q^{20} + ( -46 + 330 \beta ) q^{23} + ( -1749 + 580 \beta ) q^{25} + ( -396 + 416 \beta ) q^{26} + ( 5398 - 198 \beta ) q^{28} + ( 1094 - 1492 \beta ) q^{29} + ( -6040 + 1600 \beta ) q^{31} + ( -169 + 1729 \beta ) q^{32} + ( -6826 + 2782 \beta ) q^{34} + ( 5968 + 2528 \beta ) q^{35} + ( 454 - 2816 \beta ) q^{37} + ( -6900 + 5124 \beta ) q^{38} + ( 4360 + 1864 \beta ) q^{40} + ( 18246 - 8 \beta ) q^{41} + ( -6440 + 3112 \beta ) q^{43} + ( -2962 + 2026 \beta ) q^{46} + ( -22066 + 390 \beta ) q^{47} + ( 15709 + 9052 \beta ) q^{49} + ( -16883 + 5229 \beta ) q^{50} + ( -6996 + 524 \beta ) q^{52} + ( 2536 + 7102 \beta ) q^{53} + ( 32906 + 3974 \beta ) q^{56} + ( 19594 - 10046 \beta ) q^{58} + ( 2384 - 1980 \beta ) q^{59} + ( 13664 - 2026 \beta ) q^{61} + ( -55080 + 15640 \beta ) q^{62} + ( -19911 + 12623 \beta ) q^{64} + ( -6592 - 2796 \beta ) q^{65} + ( -13908 - 12704 \beta ) q^{67} + ( -52438 + 11614 \beta ) q^{68} + ( 21552 + 9200 \beta ) q^{70} + ( -17870 + 4354 \beta ) q^{71} + ( 26174 - 5568 \beta ) q^{73} + ( 25706 - 17350 \beta ) q^{74} + ( -88908 + 10380 \beta ) q^{76} + ( 14138 - 11426 \beta ) q^{79} + ( 1528 + 680 \beta ) q^{80} + ( 127786 - 18294 \beta ) q^{82} + ( 28740 + 21960 \beta ) q^{83} + ( -16560 - 7148 \beta ) q^{85} + ( -69976 + 25112 \beta ) q^{86} + ( 40454 - 26704 \beta ) q^{89} + ( -37880 - 9432 \beta ) q^{91} + ( -35470 + 4558 \beta ) q^{92} + ( -157582 + 24406 \beta ) q^{94} + ( -67872 - 28848 \beta ) q^{95} + ( -125746 + 9924 \beta ) q^{97} + ( 37547 + 38603 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 13q^{2} + 37q^{4} - 58q^{5} - 146q^{7} + 39q^{8} + O(q^{10}) \) \( 2q + 13q^{2} + 37q^{4} - 58q^{5} - 146q^{7} + 39q^{8} - 212q^{10} + 130q^{13} + 74q^{14} + 241q^{16} - 728q^{17} + 828q^{19} + 1072q^{20} + 238q^{23} - 2918q^{25} - 376q^{26} + 10598q^{28} + 696q^{29} - 10480q^{31} + 1391q^{32} - 10870q^{34} + 14464q^{35} - 1908q^{37} - 8676q^{38} + 10584q^{40} + 36484q^{41} - 9768q^{43} - 3898q^{46} - 43742q^{47} + 40470q^{49} - 28537q^{50} - 13468q^{52} + 12174q^{53} + 69786q^{56} + 29142q^{58} + 2788q^{59} + 25302q^{61} - 94520q^{62} - 27199q^{64} - 15980q^{65} - 40520q^{67} - 93262q^{68} + 52304q^{70} - 31386q^{71} + 46780q^{73} + 34062q^{74} - 167436q^{76} + 16850q^{79} + 3736q^{80} + 237278q^{82} + 79440q^{83} - 40268q^{85} - 114840q^{86} + 54204q^{89} - 85192q^{91} - 66382q^{92} - 290758q^{94} - 164592q^{95} - 241568q^{97} + 113697q^{98} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.37228
−2.37228
3.62772 0 −18.8397 −57.7228 0 −251.081 −184.432 0 −209.402
1.2 9.37228 0 55.8397 −0.277187 0 105.081 223.432 0 −2.59787
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.6.a.p 2
3.b odd 2 1 363.6.a.f 2
11.b odd 2 1 99.6.a.d 2
33.d even 2 1 33.6.a.e 2
132.d odd 2 1 528.6.a.o 2
165.d even 2 1 825.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.e 2 33.d even 2 1
99.6.a.d 2 11.b odd 2 1
363.6.a.f 2 3.b odd 2 1
528.6.a.o 2 132.d odd 2 1
825.6.a.c 2 165.d even 2 1
1089.6.a.p 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{2} - 13 T_{2} + 34 \)
\( T_{5}^{2} + 58 T_{5} + 16 \)
\( T_{7}^{2} + 146 T_{7} - 26384 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 34 - 13 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 + 58 T + T^{2} \)
$7$ \( -26384 + 146 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -40952 - 130 T + T^{2} \)
$17$ \( -1009172 + 728 T + T^{2} \)
$19$ \( -5817312 - 828 T + T^{2} \)
$23$ \( -884264 - 238 T + T^{2} \)
$29$ \( -18243924 - 696 T + T^{2} \)
$31$ \( 6337600 + 10480 T + T^{2} \)
$37$ \( -64511196 + 1908 T + T^{2} \)
$41$ \( 332770036 - 36484 T + T^{2} \)
$43$ \( -56044032 + 9768 T + T^{2} \)
$47$ \( 477085816 + 43742 T + T^{2} \)
$53$ \( -379065264 - 12174 T + T^{2} \)
$59$ \( -30400064 - 2788 T + T^{2} \)
$61$ \( 126184224 - 25302 T + T^{2} \)
$67$ \( -921013232 + 40520 T + T^{2} \)
$71$ \( 89872392 + 31386 T + T^{2} \)
$73$ \( 291320452 - 46780 T + T^{2} \)
$79$ \( -1006085552 - 16850 T + T^{2} \)
$83$ \( -2400814800 - 79440 T + T^{2} \)
$89$ \( -5148586428 - 54204 T + T^{2} \)
$97$ \( 13776267004 + 241568 T + T^{2} \)
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